## Nonlinear dynamic analysis of a drivetrain composed of spur, helical and spiral bevel gears

NONLINEAR DYNAMICS, vol.100, no.4, pp.3145-3170, 2020 (Journal Indexed in SCI)

• Publication Type: Article / Article
• Volume: 100 Issue: 4
• Publication Date: 2020
• Doi Number: 10.1007/s11071-020-05666-8
• Title of Journal : NONLINEAR DYNAMICS
• Page Numbers: pp.3145-3170
• Keywords: Nonlinear gear dynamics, Multi-mesh gear train, Spiral bevel gear, Time-varying mesh stiffness, Modal superposition method, Harmonic balance method with multi-harmonics, HARMONIC-BALANCE METHOD, MODAL SUPERPOSITION METHOD, VIBRATION ANALYSIS, FINITE-ELEMENTS, FORCED RESPONSE, TRANSMISSION, MODEL, OSCILLATOR, CLEARANCE, MICROSLIP

#### Abstract

This paper proposes a dynamic model for the first time in order to investigate nonlinear time-varying dynamic behavior of a drivetrain including parallel axis gears (such as spur and helical gears) and intersecting axis gears (such as spiral bevel gears). Flexibilities of shafts and bearings are included in the dynamic model by the use of finite element modeling. Finite element models of shafts are coupled with each other by the mesh models of gear pairs including backlash nonlinearity and fluctuating mesh stiffness. A system of nonlinear algebraic equations is established from the resulting nonlinear differential equations of motion by utilizing multi-harmonic harmonic balance method (HBM) in conjunction with continuous-time Fourier transform (CFT). Since the number of nonlinear equations is large, potential convergence problems are avoided by utilizing continuous-time Fourier transform, in contrast with gear dynamics studies that use discrete Fourier transform (DFT). Solutions obtained by utilizing CFT and DFT are compared, and the advantages of utilizing CFT are shown. Fourier coefficients are calculated by utilizing analytical integration rather than numerical integration for a further improvement in computational time. A new solution method, modal superposition method, is introduced for the first time to study nonlinear dynamics of drivetrains with multiple gear meshes, which is impractical if traditional solution methods are used due to the increased number of nonlinear equations. Using modal superposition method, the number of nonlinear equations becomes proportional to the number of modes employed which is significantly less than the number of degrees of freedom associated with nonlinearities, especially as the number of gear meshes in the drivetrain increases. Consequently, the proposed method decreases the computational effort drastically in the forced response analysis of multi-mesh, multi-stage gear systems and also makes it possible to model gear shafts by using finite element method. The resulting system of nonlinear algebraic equations is solved by utilizing Newton's method with arc-length continuation. Solutions obtained by HBM are validated by the solutions obtained by direct numerical integration. Several parametric studies are carried out in order to investigate the effects of design parameters on the dynamics of the drivetrain. It is observed that nonlinear modeling of helical gear pairs is necessary if they are coupled with spur or spiral bevel gear pairs.